Algebra Examples

$\frac{3 x + 5}{2 x - 1} + \frac{7 x + 3}{1 - 2 x}$

Step 1

Simplify terms.

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Step 1.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{3 x + 5}{2 x - 1} + \frac{7 x + 3}{- 1 (- 1) - 2 x}$

Step 1.2

Factor $- 1$ out of $- 2 x$ .

$\frac{3 x + 5}{2 x - 1} + \frac{7 x + 3}{- 1 (- 1) - (2 x)}$

Step 1.3

Factor $- 1$ out of $- 1 (- 1) - (2 x)$ .

$\frac{3 x + 5}{2 x - 1} + \frac{7 x + 3}{- 1 (- 1 + 2 x)}$

Step 1.4

Simplify the expression.

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Step 1.4.1

Move a negative from the denominator of $\frac{7 x + 3}{- 1 (- 1 + 2 x)}$ to the numerator.

$\frac{3 x + 5}{2 x - 1} + \frac{- (7 x + 3)}{- 1 + 2 x}$

Step 1.4.2

Reorder terms.

$\frac{3 x + 5}{2 x - 1} + \frac{- (7 x + 3)}{2 x - 1}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{3 x + 5 - (7 x + 3)}{2 x - 1}$

Step 2

Simplify the numerator.

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Step 2.1

Apply the distributive property.

$\frac{3 x + 5 - (7 x) - 1 \cdot 3}{2 x - 1}$

Step 2.2

Multiply $7$ by $- 1$ .

$\frac{3 x + 5 - 7 x - 1 \cdot 3}{2 x - 1}$

Step 2.3

Multiply $- 1$ by $3$ .

$\frac{3 x + 5 - 7 x - 3}{2 x - 1}$

Step 2.4

Subtract $7 x$ from $3 x$ .

$\frac{- 4 x + 5 - 3}{2 x - 1}$

Step 2.5

Subtract $3$ from $5$ .

$\frac{- 4 x + 2}{2 x - 1}$

Step 2.6

Factor $2$ out of $- 4 x + 2$ .

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Step 2.6.1

Factor $2$ out of $- 4 x$ .

$\frac{2 (- 2 x) + 2}{2 x - 1}$

Step 2.6.2

Factor $2$ out of $2$ .

$\frac{2 (- 2 x) + 2 (1)}{2 x - 1}$

Step 2.6.3

Factor $2$ out of $2 (- 2 x) + 2 (1)$ .

$\frac{2 (- 2 x + 1)}{2 x - 1}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $- 2 x + 1$ and $2 x - 1$ .

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Step 3.1.1

Factor $- 1$ out of $- 2 x$ .

$\frac{2 (- (2 x) + 1)}{2 x - 1}$

Step 3.1.2

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{2 (- (2 x) - 1 (- 1))}{2 x - 1}$

Step 3.1.3

Factor $- 1$ out of $- (2 x) - 1 (- 1)$ .

$\frac{2 (- (2 x - 1))}{2 x - 1}$

Step 3.1.4

Rewrite $- (2 x - 1)$ as $- 1 (2 x - 1)$ .

$\frac{2 (- 1 (2 x - 1))}{2 x - 1}$

Step 3.1.5

Cancel the common factor.

$\frac{2 (- 1 (2 x - 1))}{2 x - 1}$

Step 3.1.6

Divide $2 \cdot (- 1)$ by $1$ .

$2 \cdot (- 1)$

Step 3.2

Multiply $2$ by $- 1$ .

$- 2$